Dedekind Domain or Applications of Localization

It takes a very long time for me complete the note on Dedekind domain. We already know that the coordinate ring of a irreducible curve is a Dedekind domain if and only if our curve is smooth. The proof of this theorem is one of the purposes of the note. Furthermore, it is proved that a ring of algebraic integers over a number field is also Dedekind domain, and working with number ring, is basically, working with ideals with main tools from commutative algebra. This note is devoted to characterize the Dedekind domain, via the important concept from commutative algebra, localization. The first section is about local properties of Dedekind domain, and in the second section, we will use localization and dimension theory to prove the Hilbert's nullstellensatz for curves. Finally, we will characterize the Dedekind domain via fractional ideals.